Question

If the sum of the coefficients of all even powers of x in the product (1 + x + x² + ... + x²ⁿ) (1 - x + x² - x³ + ... + x²ⁿ) is 61, then n is equal to _________.

Answer 1

Let (1 + x + x² +…+ x²ⁿ) (1 - x + x² - x³ +…+x²ⁿ) 

= a₀ + a₁x + a₂x² + a₃x³ + a₄x⁴ +…+a_4nx⁴ⁿ

So, a₀ + a₁ + a₂ +…+ a_4n = 2n + 1 …(1)

a₀ - a₁ + a₂ - a₃…+ a_4n = 2n + 1 …(2)

⇒ a₀ + a₂ + a₄ +…+a_4n = 2n + 1

⇒ 2n + 1 = 61 ⇒ n = 30

Answer 2

Let (1 - x + x² .....) (1 + x + x² ......)

= a₀ + a₁ x + a₂x² + .........

Put x = 1

1(2n + 1) = a₀ + a₁ + a₂ + ......a_2n .....(i)

put x = -1

(2n + 1) × 1 = a₀ - a₁ + a₂ +........a_2n ........(ii)

Form (i) and (ii)

4n + 2 = 2(a₀ + a₂ +....)

= 2 × 61

⇒ 2n + 1 = 61 ⇒ n = 30